A variety of numerical analysis methods have been developed for determining the steady-state response of non-linear electronic circuits, networks and other devices. These methods include, for example, finite differencing and harmonic balance techniques. These and other steady-state analysis methods generally involve solving a boundary-value problem on a system of non-linear ordinary differential equations or differential algebraic equations. Each of the methods typically generates a large linear system of equations which needs to be solved many times during a so-called "outer" iteration of the method as it finds the solution to the system of non-linear equations. The large linear system of equations can usually be characterized as a Jacobian matrix of partial derivatives of the non-linear system, and the solution of the Jacobian matrix is a major computational bottleneck in the steady-state analysis process. For steady-state simulation of an electrical network or integrated circuit of even moderate complexity, the Jacobian matrices can be extremely difficult to solve using direct factorization.
An alternative approach to direct factorization of the Jacobian matrix involves the use of iterative linear solution methods, also referred to as iterative linear "solvers." Although Jacobian matrices are often very large, these matrices are typically structured in a manner which facilitates fast matrix-vector multiplication, and can therefore be solved using well-known iterative linear solution methods such as the QMR or GMRES algorithms. The speed and robustness of these and other iterative linear methods can depend critically on the choice of a preconditioner, which is used to generate a relatively easy-to-invert approximation to a given Jacobian matrix. Unfortunately, conventional preconditioning techniques provide insufficient reductions in computational complexity. As a result, steady-state analysis using iterative linear solution methods remains unduly computationally intensive and therefore impractical in a wide variety of important applications.